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Properties

Label 2-3240-9.4-c1-0-34
Degree $2$
Conductor $3240$
Sign $0.173 + 0.984i$
Analytic cond. $25.8715$
Root an. cond. $5.08640$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + (1 + 1.73i)11-s + (2.5 − 4.33i)13-s − 4·17-s − 5·19-s + (1 − 1.73i)23-s + (−0.499 − 0.866i)25-s + (−5 − 8.66i)29-s + (4 − 6.92i)31-s − 0.999·35-s − 3·37-s + (−3 + 5.19i)41-s + (−2 − 3.46i)43-s + (4 + 6.92i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.188 + 0.327i)7-s + (0.301 + 0.522i)11-s + (0.693 − 1.20i)13-s − 0.970·17-s − 1.14·19-s + (0.208 − 0.361i)23-s + (−0.0999 − 0.173i)25-s + (−0.928 − 1.60i)29-s + (0.718 − 1.24i)31-s − 0.169·35-s − 0.493·37-s + (−0.468 + 0.811i)41-s + (−0.304 − 0.528i)43-s + (0.583 + 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(25.8715\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.211392381\)
\(L(\frac12)\) \(\approx\) \(1.211392381\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5 + 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 9T + 73T^{2} \)
79 \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.376409790926206178647905351340, −7.891678216095794416998142827967, −6.93410475560692207718248810314, −6.24108497250037474737755153225, −5.57640740447330907711964446747, −4.45418684630574356493718713006, −3.89230188942959940168501655324, −2.72658674335477858753098944222, −1.96939720214591320692318070281, −0.38934661454709129745777601041, 1.18961209687811799822285303567, 2.10983549793916148648753275501, 3.48797558544588749895944457065, 4.11724954176897725403269792128, 4.89094855094810274813565626061, 5.80410108684647222389025687379, 6.81546060984119447939551891464, 7.04459811558493876626876272795, 8.398943981190422521671878335549, 8.722733822840764520888617058245

Graph of the $Z$-function along the critical line